Generally it's not ideal to have music playing in your runs because it can make it harder for mods to check if the run is spliced and such, game audio is an important factor in verifying runs, there's some way you can record the game without your music playing in the recording but I forget what's used for that
I use OBS to record and stream if that helps. I was thinking of letting the viewers add songs as I go on
I haven't seen a case of a run being rejected because they used their own music. Check with the game moderators if you're unsure.
I've heard of moderators saying they would reject it for music, example is that thread a bit back with the resident evil mods talking about music in RE runs, the run was rejected due to music
RE is the main example of runs getting rejected for having music. Just check with the mod and if they allow it. Tends to be a case-by-case thing.